2 (number)

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2 (two) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3.

In mathematics

Two has many properties in mathematics.^[1] An integer is called even if it is divisible by 2. For integers written in a numeral system based on an even number, such as decimal and hexadecimal, divisibility by 2 is easily tested by merely looking at the one's place digit. If it's even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8.

Two is the smallest and the first prime number, and the only even one (for this reason it is sometimes humorously called "the oddest prime"). The next prime is three. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, and the first Smarandache-Wellin prime. It is an Eisenstein prime with no imaginary part and real part of the form Failed to parse (Missing texvc executable; please see math/README to configure.): 3n - 1 . It is also a Stern prime, a Pell number, and a Markov number, appearing in infinitely many solutions to the Markov Diophantine equation involving odd-indexed Pell numbers.

It is the third Fibonacci number, and the third and fifth Perrin numbers.

Despite being a prime, two is also a highly composite number, because it has more divisors than the number one. The next highly composite number is four.

Vulgar fractions with 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with most primes, because 2 and 5 are factors of ten, the decimal base.

Two is the base of the simplest numeral system in which natural numbers can be written concisely, being the length of the number a logarithm of the value of the number (whereas in base 1 the length of the number is the value of the number itself); the binary system is used in computers.

For any number x:

x+x = 2·x addition to multiplication

x·x = x² multiplication to exponentiation

x^x = x↑↑2 exponentiation to tetration

Two also has the unique property that 2+2 = 2·2 = 2²=2↑↑2=2↑↑↑2, and so on, no matter how high the operation is.

Two is the only number x such that the sum of the reciprocals of the powers of x equals itself. In symbols: Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=0}^{\infin}\frac {1}{2^k}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=2

This comes from the fact that:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=0}^\infin \frac {1}{n^k}=1+\frac{1}{n-1} \quad\mbox{for all} \quad n\in\mathbb R > 1

Powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent.

Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the exponent would normally be written for cubic roots and other such roots, is left blank for square roots, as it is considered tacit.

The square root of two was the first known irrational number.

The smallest field has two elements.

In the set-theoretical construction of the natural numbers, 2 is identified with the set Failed to parse (Missing texvc executable; please see math/README to configure.): \{\{\emptyset\},\emptyset\} . This latter set is important in category theory: it is a subobject classifier in the category of sets.

Two is a primorial, as well as its own factorial. Two often occurs in numerical sequences, such as the Fibonacci number sequence, but not quite as often as one does. Two is also a Motzkin number, a Bell number, an all-Harshad number, a meandric number, a semi-meandric number, and an open meandric number.

Two is the number of n-Queens Problem solutions for n = 4. With one exception, all known solutions to Znám's problem start with 2.

Two also has the unique property such that:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=0}^{n-1} 2^k = 2^{n} - 1

and also

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=a}^{n-1} 2^k = 2^n - \sum_{k=0}^{a-1} 2^k - 1

for a not equal to zero

Two has a connection to triangular numbers:

Failed to parse (Missing texvc executable; please see math/README to configure.): \prod_{k=0}^n 2^k= 2^{tri_2(n)}

Where Failed to parse (Missing texvc executable; please see math/README to configure.): tri_d(n)= \frac {1}{d!}\prod_{k=0}^{d-1} (n+k)\quad \mbox{if}\quad d\ge 2

gives the nth d-dimensional simplex number. When d=2, 

Failed to parse (Missing texvc executable; please see math/README to configure.): tri_2(n)=\frac {n^2+n}{2}

The number of domino tilings of a 2×2 checkerboard is 2.

For any polyhedron homeomorphic to a sphere, the Euler characteristic is Failed to parse (Missing texvc executable; please see math/README to configure.): \chi = V-E+F = 2.

As of 2008, there are only two known Wieferich primes.

List of basic calculations

Evolution of the glyph

The glyph we use today in the Western world to represent the number 2 traces its roots back to the Brahmin Indians, who wrote 2 as two horizontal lines (it is still written that way in modern Chinese and Japanese). The Gupta rotated the two lines 45 degrees, making them diagonal, and sometimes also made the top line shorter and made its bottom end curve towards the center of the bottom line. Apparently for speed, the Nagari started making the top line more like a curve and connecting to the bottom line. The Ghubar Arabs made the bottom line completely vertical, and now the glyph looked like a dotless closing question mark. Restoring the bottom line to its original horizontal position, but keeping the top line as a curve that connects to the bottom line leads to our modern glyph.^[2]

In fonts with text figures, 2 usually has the same height as a lowercase X, for example, Image:Text figures 256.svg.

In science

• The number of polynucleotide strands in a DNA double helix.
• The first magic number.^[3]
  • The atomic number of helium.
  • Group 2 in the Periodic table of the elements consists of the alkaline earth metals whose usual valence is +2.
  • Period 2 in the Periodic table consists of the eight elements lithium through neon.

  Astronomy

  • Messier object M2, a magnitude 6.5 globular cluster in the constellation Aquarius.
  • The New General Catalogue object NGC 2, a magnitude 14.2 spiral galaxy in the constellation Pegasus
  • The Saros number of the solar eclipse series which began on May 4 2861 BC and ended on June 21 1563 BC . The duration of Saros series 2 was 1298.1 years, and it contained 73 solar eclipses.
  • The Saros number of the lunar eclipse series which began on February 21 2541 BC and ended on April 22 1225 BC. The duration of Saros series 2 was 1316.2 years, and it contained 74 lunar eclipses.
  • The Roman numeral II stands for bright giant in the Yerkes spectral classification scheme.
  • The Roman numeral II (usually) stands for the second-discovered satellite of a planet or minor planet (e.g. Pluto II or (87) Sylvia II Remus)
  • A binary star is a stellar system consisting of two stars orbiting around their center of mass.

  In religion

  • The Ten Commandments were given in the form of two tablets (Shnei Luchot HaBrit)
  • Two candles are traditionally kindled to usher in the Shabbat, recalling the two different ways Shabbat is referred to in the two times the Ten Commandments are recorded in the Torah. These two expressions are known in Hebrew as שמור וזכור ("guard" and "remember"), as in "Guard the Shabbat day to sanctify it" (Deut. 5:12) and "Remember the Shabbat day to sanctify it" (Ex. 20:8)
  • Two challahs (lechem mishnah) are placed on the table for each Shabbat meal and a blessing made over them, to commemorate the double portion of manna which fell in the desert every Friday to cover that day's meals and the Shabbat meals
  • In Jewish law, the testimony of two witnesses are required to verify and validate events, such as marriage, divorce, and a crime that warrants capital punishment
  • Rosh Hashana, the first day of the Jewish year, is a 2-day holiday
  • "Second-Day Yom Tov" (Yom Tov Sheini Shebegaliyot) is a rabbinical enactment that mandates a two-day celebration for each of the one-day Jewish festivals (i.e., the first and seventh day of Passover, the day of Shavuot, the first day of Sukkot, and the day of Shemini Atzeret) outside the land of Israel
  • Animals boarded Noah's Ark two by two.

  In culture

  The most common philosophical dichotomy is perhaps the one of good and evil, but there are many others. See dualism for an overview. In Hegelian dialectic, the process of synthesis creates two perspectives from one.

  Two (二, èr) is a good number in Chinese culture. There is a Chinese saying "good things come in pairs". It is common to use double symbols in product brandnames, e.g. double happiness, double coin, double elephants etc. Cantonese people like the number two because it sounds the same as the word "easy" (易) in Cantonese.

  In Finland, two candles are lit on Independence Day. Putting them on the windowsill invokes the symbolical meaning of division, and thus independence.

  The numeral 2 is sometimes used as a shorthand for "to", such as b2b (business-to-business). This form of abbreviation is especially common in the world of popular music, hip-hop, rap, and R&B, where the "too" or "to" in group names, album titles, and song names is replaced with "2". The Eurodance group 2 Unlimited is one example. In some cases, the Roman numeral II is substituted, as seen in the name of R&B group Boyz II Men. The names of computer software that translate data from one format into another also commonly employ the numeral 2, such as dvi2ps and texi2roff.

  In pre-1972 Indonesian and Malay orthography, 2 was shorthand for the reduplication that forms plurals: orang "person", orang-orang or orang2 "people".

  Among children, or when otherwise calling for subtlety, the phrase "number 2" can refer to the act of defecating (with urination being "number 1").

  In sports

  In American football, 2 is a jersey number that can be worn by quarterbacks, kickers, and punters.

  In American football, a team can achieve 2 points via a two-point conversion after scoring a touchdown or via a safety.

  In baseball, two represents the catcher's position.

  2 is the retired number for former baseball great Charlie Gehringer.

  2 was the number used by Leo Durocher as baseball player, coach, and manager from the early 1930s until his retirement.

  In football, number 2 is assigned in most cases to the right defender or fullback

  In basketball terminology, two represents the shooting guard or "off guard".

  In basketball, 2 points are awarded for a field goal.

  In cricket, 2 is the number of members of the batting side of the field of play at any one time.

  From 1991 until his retirement in 2005, 2 was the number borne on the car of NASCAR Nextel Cup driver Rusty Wallace. It was also the number of the car which Dale Earnhardt drove to the first of his seven season championships in 1980. Kurt Busch, as of 2006 drives the 2 car.

  In rugby league, 2 is the jersey number of the starting right wing. It is also the number of points received for a successful conversion after a try is scored or a successful penalty kick.

  In rugby union, 2 is the jersey number of the starting hooker. It is also the number of points received for a successful conversion after a try is scored.

  In rowing, 2- refers to a coxless pair; 2+ refers to a coxed pair; 2x refers to a coxless double.

  In technology

  Two is the DVD region in Europe, South Africa, the Middle East and Japan.

  Two is the first digit of international telephone dialing codes primarily for countries in Africa. On most phones, the 2 key is associated with the letters A, B, and C.

  On American computer keyboards, the "2" key is associated with the "@" symbol. On British computer keyboards, the "2" key is associated with the " symbol.

  The "two meter band" in amateur radio includes the frequencies from 144 to 146 MHz. In the Americas, the two meter band extends to 148 MHz.

  In other fields

  

  

  International maritime signal flag for 2

  Groups of two:

  • Lists of pairs
    • list of twins

  Two is:

  • The denomination of U.S. paper currency with Thomas Jefferson's portrait on it; last issued 2003; as of 2006 rarely seen in general circulation.
  • The denomination of the Canadian monetary coin featuring a polar bear, commonly known as the "toonie" after the $1 coin, the loonie.
  • The denomination of the $2 New Zealand coin, featuring the white heron.
  • The name of several fictional characters: Number Two.
  • The designation of the Trans-Canada Highway in most of the province of New Brunswick.
  • U.S. Route 2, two separated highways in the northern tier of the United States, the western segment connecting Everett, Washington to St. Ignace, Michigan and the eastern route connecting Rouses Point, New York to Houlton, Maine.
  • The lowest channel of television in the United States, Canada, and Mexico on which television signals are broadcast.
  • One of the radio callsign prefixes allocated to the United Kingdom.
  • In Astrology, Taurus is the 2nd sign of the Zodiac.

  References

  1. ^ Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 41–44
  2. ^ Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 393, Fig. 24.62
  3. ^ Xavier Borg, Engineer (February 2, 2006). Magic numbers derived from VPM nuclear model. Blaze Labs Research. Retrieved on 2007-03-24.

  
  

  

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